Answer to Question #164897 in Quantum Mechanics for Prem

Question

 Quantum mechanics is relevant, when the de Broglie wavelength of the particle is 

greater than the distance between particles. The purpose of this problem is to determine 

which systems will have to be treated quantum mechanically and which can be 

described classically.

a) Show that the typical de Broglie wavelength of a particle in an ideal gas in 

equilibrium is 𝜆 =

ℎ

√3𝑚𝐾𝐵𝑇

Answer 1

a) In thermal equilibrium at (Kelvin) temperature T, the average kinetic energy of a particle is

$$\frac{p^2}{2m}=\frac{3}{2}kT$$

so the typical de Broglie wavelength is

$$\lambda=\frac{h}{\sqrt{3mkT}}$$

b)

$$T=\frac{h^2}{3mkd^2}\\T=\frac{(6.6\cdot10^{-34})^2}{3(3.9\cdot10^{-26})(1.4\cdot10^{-23})(3\cdot10^{-10})^2}=3\ K$$

c)

$$Pd^3=kT\\T=\frac{1}{k}\left(\frac{h^2}{3m}\right)^\frac{3}{5}(P)^\frac{2}{5}$$

He:

$$T=\frac{1}{(1.4\cdot10^{-23})}\left(\frac{(6.6\cdot10^{-34})^2}{3(6.8\cdot10^{-27})}\right)^\frac{3}{5}(10^5)^\frac{2}{5}=2.8\ K$$

To treat the helium quantum mechanically we need it to be in a temperature less than 2.8 K.

H:

$$T=\frac{h^2}{3mkd^2}\\T=\frac{(6.6\cdot10^{-34})^2}{3(3.4\cdot10^{-27})(1.4\cdot10^{-23})(0.01)^2}=3.1\cdot10^{-14}\ K$$

In the outer space hydrogen shows a classical behaviour.